Problem: Divide the polynomials.
Solution: Usually, there are many different ways to divide polynomials. Here, we will use the method of polynomial long division. Notice the numerator is missing a $1^{\text{st}}$ degree term. Let's add it as $0x$. $\begin{array}{r} x-\phantom{1}4 \\ x+4|\overline{x^2+0x+\phantom{6}1} \\ \mathllap{-(}\underline{x^2+4x\phantom{+16}\rlap )} \\ -4x+\phantom{6}1 \\ \mathllap{-(}\underline{-4x-16\rlap )} \\ 17 \end{array}$ We get that the quotient is $x-4$ and the remainder is $17$, and therefore: $\dfrac{x^2+1}{x+4}=x-4+\dfrac{17}{x+4}$ [I want to see a different way of performing the division.]